# A class of exponential neighbourhoods for the quadratic travelling salesman problem

## TL;DR

This paper investigates exponential neighborhoods for the quadratic TSP, introducing fixed-rank variants, analyzing their complexity, and providing algorithms with approximation schemes for specific cases, expanding understanding of solution approaches.

## Contribution

It introduces the fixed-rank QTSP, analyzes its complexity, and develops algorithms with FPTAS for certain cases, advancing solution methods for quadratic TSP variants.

## Key findings

- Fixed-rank QTSP solvable in pseudopolynomial time.
- FPTAS exists for most fixed-rank QTSP cases.
- Matching edge ejection tours are NP-hard in the quadratic case.

## Abstract

The Quadratic Travelling Salesman Problem (QTSP) is to find a least-cost Hamiltonian cycle in an edge-weighted graph, where costs are defined on all pairs of edges such that each edge in the pair is contained in the Hamiltonian cycle. This is a more general version than the one that appears in the literature as the QTSP, denoted here as the \emph{adjacent quadratic} TSP, which only considers costs for pairs of adjacent edges. Major directions of research work on the linear TSP include exact algorithms, heuristics, approximation algorithms, polynomially solvable special cases and exponential neighbourhoods among others. In this paper we explore the complexity of searching exponential neighbourhoods for QTSP, the fixed-rank QTSP, and the adjacent quadratic TSP. The fixed-rank QTSP is introduced as a restricted version of the QTSP where the cost matrix has fixed rank $p$. It is shown that fixed-rank QTSP is solvable in pseudopolynomial time and admits an FPTAS for each of the special cases studied, except for the case of matching edge ejection tours. The adjacent quadratic TSP is shown to be polynomially-solvable in many of the cases for which the linear TSP is polynomially-solvable. Interestingly, optimizing over the matching edge ejection tour neighbourhood is shown to be pseudopolynomial for the rank 1 case without a linear term in the objective function, but NP-hard for the adjacent quadratic TSP case. We also show that the quadratic shortest path problem on an acyclic digraph can be solved in pseudopolynomial time and by an FPTAS when the rank of the associated cost matrix is fixed.